Find The Sum, $S_n$, For The Arithmetic Series Described. Use The Formula: $S_n=\frac{n}{2}\left(a_1+a_n\right)$Where: - \$a_1 = 8$[/tex\]- $a_n = 73$- $n = 11$a. \$S_n =

Understanding the Arithmetic Series Formula

In mathematics, an arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. The formula for finding the sum of an arithmetic series is given by:

$$S_n=\frac{n}{2}\left(a_1+a_n\right)$$

where:

• $S_n$ is the sum of the first $n$ terms of the series
• $n$ is the number of terms in the series
• $a_1$ is the first term of the series
• $a_n$ is the last term of the series

Given Values

We are given the following values for the arithmetic series:

• $a_1 = 8$ (the first term of the series)
• $a_n = 73$ (the last term of the series)
• $n = 11$ (the number of terms in the series)

Finding the Sum

To find the sum of the arithmetic series, we can plug the given values into the formula:

$$S_n=\frac{n}{2}\left(a_1+a_n\right)$$

Substituting the given values, we get:

$$S_n=\frac{11}{2}\left(8+73\right)$$

Simplifying the expression, we get:

$$S_n=\frac{11}{2}\left(81\right)$$

$$S_n=\frac{891}{2}$$

$$S_n=445.5$$

Therefore, the sum of the arithmetic series is $445.5$.

Conclusion

In this article, we used the formula for finding the sum of an arithmetic series to calculate the sum of a given series. We plugged in the given values and simplified the expression to find the sum. The sum of the arithmetic series is $445.5$.

Arithmetic Series Formula

The formula for finding the sum of an arithmetic series is given by:

$$S_n=\frac{n}{2}\left(a_1+a_n\right)$$

where:

• $S_n$ is the sum of the first $n$ terms of the series
• $n$ is the number of terms in the series
• $a_1$ is the first term of the series
• $a_n$ is the last term of the series

Example Use Case

Suppose we have an arithmetic series with the following values:

• $a_1 = 2$ (the first term of the series)
• $a_n = 27$ (the last term of the series)
• $n = 10$ (the number of terms in the series)

We can use the formula to find the sum of the series:

$$S_n=\frac{n}{2}\left(a_1+a_n\right)$$

Substituting the given values, we get:

$$S_n=\frac{10}{2}\left(2+27\right)$$

Simplifying the expression, we get:

$$S_n=\frac{10}{2}\left(29\right)$$

$$S_n=\frac{290}{2}$$

$$S_n=145$$

Therefore, the sum of the arithmetic series is $145$.

Real-World Applications

Arithmetic series have many real-world applications, such as:

• Finance: Arithmetic series can be used to calculate the total amount of money that will be paid out over a period of time, such as the total amount of interest paid on a loan.
• Science: Arithmetic series can be used to calculate the total amount of energy released by a series of explosions, such as a nuclear reaction.
• Engineering: Arithmetic series can be used to calculate the total amount of stress on a structure, such as a bridge.

Conclusion

In conclusion, arithmetic series are a fundamental concept in mathematics that have many real-world applications. The formula for finding the sum of an arithmetic series is given by:

$$S_n=\frac{n}{2}\left(a_1+a_n\right)$$

where:

• $S_n$ is the sum of the first $n$ terms of the series
• $n$ is the number of terms in the series
• $a_1$ is the first term of the series
• $a_n$ is the last term of the series

Frequently Asked Questions

Q: What is an arithmetic series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant.

Q: What is the formula for finding the sum of an arithmetic series?

A: The formula for finding the sum of an arithmetic series is given by:

$$S_n=\frac{n}{2}\left(a_1+a_n\right)$$

where:

• $S_n$ is the sum of the first $n$ terms of the series
• $n$ is the number of terms in the series
• $a_1$ is the first term of the series
• $a_n$ is the last term of the series

Q: How do I use the formula to find the sum of an arithmetic series?

A: To use the formula, simply plug in the given values for $a_1$, $a_n$, and $n$ into the formula and simplify the expression.

Q: What are some real-world applications of arithmetic series?

A: Arithmetic series have many real-world applications, such as:

• Finance: Arithmetic series can be used to calculate the total amount of money that will be paid out over a period of time, such as the total amount of interest paid on a loan.
• Science: Arithmetic series can be used to calculate the total amount of energy released by a series of explosions, such as a nuclear reaction.
• Engineering: Arithmetic series can be used to calculate the total amount of stress on a structure, such as a bridge.

Q: Can I use the formula to find the sum of a finite arithmetic series?

A: Yes, the formula can be used to find the sum of a finite arithmetic series. Simply plug in the given values for $a_1$, $a_n$, and $n$ into the formula and simplify the expression.

Q: Can I use the formula to find the sum of an infinite arithmetic series?

A: No, the formula cannot be used to find the sum of an infinite arithmetic series. The formula is only applicable to finite arithmetic series.

Q: What is the difference between an arithmetic series and a geometric series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant, while a geometric series is a sequence of numbers in which the ratio between any two consecutive terms is constant.

Q: Can I use the formula to find the sum of a mixed arithmetic-geometric series?

A: No, the formula cannot be used to find the sum of a mixed arithmetic-geometric series. The formula is only applicable to arithmetic series.

Q: What are some common mistakes to avoid when using the formula?

A: Some common mistakes to avoid when using the formula include:

• Incorrectly plugging in values: Make sure to plug in the correct values for $a_1$, $a_n$, and $n$ into the formula.
• Not simplifying the expression: Make sure to simplify the expression after plugging in the values.
• Using the formula for an infinite series: The formula is only applicable to finite arithmetic series.

Conclusion

In conclusion, arithmetic series are a fundamental concept in mathematics that have many real-world applications. The formula for finding the sum of an arithmetic series is given by:

$$S_n=\frac{n}{2}\left(a_1+a_n\right)$$

where:

• $S_n$ is the sum of the first $n$ terms of the series
• $n$ is the number of terms in the series
• $a_1$ is the first term of the series
• $a_n$ is the last term of the series

By using this formula, we can calculate the sum of an arithmetic series and apply it to real-world problems.